Scale vs Conformal invariance from Entanglement Entropy

TL;DR

This paper investigates the relationship between scale invariance and conformal invariance in four-dimensional theories using entanglement entropy, showing that certain conditions imply a theory is conformal or fixes coupling coefficients.

Contribution

It demonstrates that positive entanglement entropy universal parts imply scale invariance leads to conformal invariance or constrains coupling constants in 4D theories.

Findings

Positive entanglement entropy implies conformal invariance in absence of certain operators.

Presence of a scalar operator fixes the nonlinear coupling coefficient to a conformal value.

The analysis links entanglement entropy properties to fundamental symmetry structures in quantum field theories.

Abstract

For a generic conformal field theory (CFT) in four dimensions, the scale anomaly dictates that the universal part of entanglement entropy across a sphere ($\mathcal{C}_{\text{univ}}(S^{2})$) is positive. Based on this fact, we explore the consequences of assuming positive sign for $\mathcal{C}_{\text{univ}}(S^{2})$ in a four dimensional scale invariant theory (SFT). In absence of a dimension two scalar operator $\mathcal{O}_{2}$ in the spectrum of a SFT, we show that this assumption suggests that SFT is a CFT. In presence of $\mathcal{O}_{2}$, we show that this assumption can fix the coefficient of the nonlinear coupling term $\int\hspace{-.5mm} d^{4}x\sqrt{g} R\mathcal{O}_{2}$ to a conformal value.